Integrand size = 44, antiderivative size = 142 \[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\frac {x \left (11 b^2+2 a x^2\right )}{3 \sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (5 b^3+2 a b x^2\right )}{3 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {5 b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
1/3*x*(2*a*x^2+11*b^2)/(a*x^2+b^2)^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2)+2/3*x *(2*a*b*x^2+5*b^3)/(a*x^2+b^2)/(b+(a*x^2+b^2)^(1/2))^(1/2)-5*b^(3/2)*arcta n(a^(1/2)*x/b^(1/2)/(b+(a*x^2+b^2)^(1/2))^(1/2))/a^(1/2)
Time = 0.34 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\frac {x \left (11 b^2+2 a x^2\right )}{3 \sqrt {b^2+a x^2} \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 x \left (5 b^3+2 a b x^2\right )}{3 \left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {5 b^{3/2} \arctan \left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \]
(x*(11*b^2 + 2*a*x^2))/(3*Sqrt[b^2 + a*x^2]*Sqrt[b + Sqrt[b^2 + a*x^2]]) + (2*x*(5*b^3 + 2*a*b*x^2))/(3*(b^2 + a*x^2)*Sqrt[b + Sqrt[b^2 + a*x^2]]) - (5*b^(3/2)*ArcTan[(Sqrt[a]*x)/(Sqrt[b]*Sqrt[b + Sqrt[b^2 + a*x^2]])])/Sqr t[a]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a x^2-b^2\right )^2 \sqrt {\sqrt {a x^2+b^2}+b}}{\left (a x^2+b^2\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {4 b^2 \sqrt {\sqrt {a x^2+b^2}+b}}{a x^2+b^2}+\sqrt {\sqrt {a x^2+b^2}+b}+\frac {4 b^4 \sqrt {\sqrt {a x^2+b^2}+b}}{\left (a x^2+b^2\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b-\sqrt {-a} x}dx-b \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{b+\sqrt {-a} x}dx-a b^2 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b-a x\right )^2}dx-a b^2 \int \frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\left (\sqrt {-a} b+a x\right )^2}dx+\frac {2 b x}{\sqrt {\sqrt {a x^2+b^2}+b}}+\frac {2 a x^3}{3 \left (\sqrt {a x^2+b^2}+b\right )^{3/2}}\) |
3.21.12.3.1 Defintions of rubi rules used
\[\int \frac {\left (a \,x^{2}-b^{2}\right )^{2} \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}{\left (a \,x^{2}+b^{2}\right )^{2}}d x\]
Timed out. \[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int \frac {\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} - b^{2}\right )^{2}}{\left (a x^{2} + b^{2}\right )^{2}}\, dx \]
\[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int { \frac {{\left (a x^{2} - b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{2}} \,d x } \]
\[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int { \frac {{\left (a x^{2} - b^{2}\right )}^{2} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}{{\left (a x^{2} + b^{2}\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (-b^2+a x^2\right )^2 \sqrt {b+\sqrt {b^2+a x^2}}}{\left (b^2+a x^2\right )^2} \, dx=\int \frac {{\left (a\,x^2-b^2\right )}^2\,\sqrt {b+\sqrt {b^2+a\,x^2}}}{{\left (b^2+a\,x^2\right )}^2} \,d x \]